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Properties

Label 2-4304-1.1-c1-0-118
Degree 22
Conductor 43044304
Sign −1-1
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.92·3-s − 2.54·5-s − 2.70·7-s + 5.57·9-s − 2.51·11-s + 1.02·13-s − 7.45·15-s − 0.352·17-s + 7.25·19-s − 7.91·21-s − 2.93·23-s + 1.48·25-s + 7.55·27-s + 1.15·29-s − 3.02·31-s − 7.35·33-s + 6.88·35-s − 8.69·37-s + 2.98·39-s + 1.09·41-s − 4.12·43-s − 14.2·45-s + 2.15·47-s + 0.308·49-s − 1.03·51-s − 2.66·53-s + 6.39·55-s + ⋯
L(s)  = 1  + 1.69·3-s − 1.13·5-s − 1.02·7-s + 1.85·9-s − 0.756·11-s + 0.283·13-s − 1.92·15-s − 0.0855·17-s + 1.66·19-s − 1.72·21-s − 0.611·23-s + 0.297·25-s + 1.45·27-s + 0.214·29-s − 0.542·31-s − 1.28·33-s + 1.16·35-s − 1.43·37-s + 0.478·39-s + 0.170·41-s − 0.629·43-s − 2.11·45-s + 0.314·47-s + 0.0440·49-s − 0.144·51-s − 0.365·53-s + 0.862·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: −1-1
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4304, ( :1/2), −1)(2,\ 4304,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
269 1+T 1 + T
good3 1−2.92T+3T2 1 - 2.92T + 3T^{2}
5 1+2.54T+5T2 1 + 2.54T + 5T^{2}
7 1+2.70T+7T2 1 + 2.70T + 7T^{2}
11 1+2.51T+11T2 1 + 2.51T + 11T^{2}
13 1−1.02T+13T2 1 - 1.02T + 13T^{2}
17 1+0.352T+17T2 1 + 0.352T + 17T^{2}
19 1−7.25T+19T2 1 - 7.25T + 19T^{2}
23 1+2.93T+23T2 1 + 2.93T + 23T^{2}
29 1−1.15T+29T2 1 - 1.15T + 29T^{2}
31 1+3.02T+31T2 1 + 3.02T + 31T^{2}
37 1+8.69T+37T2 1 + 8.69T + 37T^{2}
41 1−1.09T+41T2 1 - 1.09T + 41T^{2}
43 1+4.12T+43T2 1 + 4.12T + 43T^{2}
47 1−2.15T+47T2 1 - 2.15T + 47T^{2}
53 1+2.66T+53T2 1 + 2.66T + 53T^{2}
59 1+7.47T+59T2 1 + 7.47T + 59T^{2}
61 1+14.2T+61T2 1 + 14.2T + 61T^{2}
67 1+6.89T+67T2 1 + 6.89T + 67T^{2}
71 1−3.29T+71T2 1 - 3.29T + 71T^{2}
73 1−0.0902T+73T2 1 - 0.0902T + 73T^{2}
79 1+11.7T+79T2 1 + 11.7T + 79T^{2}
83 1+7.80T+83T2 1 + 7.80T + 83T^{2}
89 1−0.397T+89T2 1 - 0.397T + 89T^{2}
97 1−1.05T+97T2 1 - 1.05T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.945198322818822834328045353082, −7.53928786585043002108823535383, −6.95456727924747600338435430983, −5.85593420328570223968053493212, −4.76406254002509312902180381018, −3.84177057707705627819759717268, −3.28051069539121598667366424038, −2.85603906964961265469184350587, −1.60894085843803229738741102040, 0, 1.60894085843803229738741102040, 2.85603906964961265469184350587, 3.28051069539121598667366424038, 3.84177057707705627819759717268, 4.76406254002509312902180381018, 5.85593420328570223968053493212, 6.95456727924747600338435430983, 7.53928786585043002108823535383, 7.945198322818822834328045353082

Graph of the ZZ-function along the critical line